One option might be to use the pusher module (https://marketplace.mendix.com/link/component/107957/Mendix/Pusher
This allows you to create a subscription to messages and notify users of the changes and can be customized to fit your needs.
What if there is a Pluggable widget which subscribes to Firebase FCM(it could be some other Push Notification Service Provider)
When the widget Loads we could load the Firebase Configs and then subscribe to receive messages (before that ask permission from user to receive notifications)
When a message comes then we could trigger a MF which creates and commits to a Entity which you can do by enabling system properties createdon. Which then can be used to drive your icon.
Seems like a long shot but possible.